Asymptotics for the survival probability in a killed branching random walk

TL;DR

This paper analyzes the asymptotic decay of the probability that a supercritical branching random walk maintains an infinite path above a line with slope close to its asymptotic speed, revealing a precise exponential decay rate.

Contribution

It establishes the asymptotic decay rate of survival probability in a killed branching random walk as the threshold approaches the asymptotic speed, answering an open question for Bernoulli cases.

Findings

Probability decays like exp(-beta/epsilon^{1/2}) as epsilon approaches zero.

Provides a general asymptotic result under mild assumptions.

Answers an open question for Bernoulli(p) branching processes.

Abstract

Consider a discrete-time one-dimensional supercritical branching random walk. We study the probability that there exists an infinite ray in the branching random walk that always lies above the line of slope $\gamma-\epsilon$, where $\gamma$ denotes the asymptotic speed of the right-most position in the branching random walk. Under mild general assumptions upon the distribution of the branching random walk, we prove that when $\epsilon\to 0$, the probability in question decays like $\exp\{- {\beta + o(1)\over \epsilon^{1/2}}\}$, where $\beta$ is a positive constant depending on the distribution of the branching random walk. In the special case of i.i.d. Bernoulli$(p)$ random variables (with $0<p<{1\over 2}$) assigned on a rooted binary tree, this answers an open question of Robin Pemantle.